document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209745>Question 1209745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks, to complete the factorization:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850081 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850081\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to factor the expression:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Expand the left side:**<BR>\n" );
document.write( "   (a² + b² - c²)² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Rearrange the terms strategically:**  We want to try to create perfect squares. Notice the 4a²bc term.  Let's try to incorporate it with other terms:\r<BR>\n" );
document.write( "\n" );
document.write( "   a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² + 2a²c² + 2b²c² - 4a²c² - 4b²c² + 4a²bc\r<BR>\n" );
document.write( "\n" );
document.write( "   = a⁴ + b⁴ + c⁴ + 2a²b² + 2a²c² + 2b²c² - 4a²c² - 4b²c² + 4a²bc<BR>\n" );
document.write( "   = (a² + b² + c²)² - 4c²(a² + b² - ab)\r<BR>\n" );
document.write( "\n" );
document.write( "   This doesn't seem to be getting us anywhere useful. Let's try a different approach.\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Try grouping terms differently:**<BR>\n" );
document.write( "    a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc = a⁴ + 2a²b² + b⁴ - 2a²c² + 4a²bc - 2b²c² + c⁴<BR>\n" );
document.write( "    = (a² + b²)² - 2c²(a² - 2abc/2c² + b²) + c⁴<BR>\n" );
document.write( "    = (a² + b²)² - 2c²(a² + b² - 2abc/2c²) + c⁴\r<BR>\n" );
document.write( "\n" );
document.write( "Let's try another approach.  We can rewrite the expression as:\r<BR>\n" );
document.write( "\n" );
document.write( "(a² + b² - c²)² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc\r<BR>\n" );
document.write( "\n" );
document.write( "Notice if we rewrite 4a²bc as 2a²bc + 2a²bc, we can rearrange:\r<BR>\n" );
document.write( "\n" );
document.write( "= a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 2a²bc + 2a²bc<BR>\n" );
document.write( "= a⁴ + b⁴ + c⁴ + 2a²b² + 2a²bc - 2a²c² + 2a²bc - 2b²c²<BR>\n" );
document.write( "= a⁴ + b⁴ + c⁴ + 2a²(b² + bc - c²) + 2a²bc - 2b²c²<BR>\n" );
document.write( "= (a² + b² + c² + 2ab)(a² + b² + c² - 2ac)(a² + b² + c² - 2bc)\r<BR>\n" );
document.write( "\n" );
document.write( "This also doesn't immediately lead to the desired factorization.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's look at the original expression again: (a² + b² - c²)² + 4a²bc\r<BR>\n" );
document.write( "\n" );
document.write( "If we let c = 0, we get (a² + b²)² which factors into (a + ib)(a - ib)(a + ib)(a - ib).  This doesn't seem to help much either.\r<BR>\n" );
document.write( "\n" );
document.write( "After trying several approaches, the given expression doesn't seem to readily factor into the form (a + ...)(a + ...)(a + ...)(a + ...). It's possible there's a typo in the original problem, or it might require a more complex factorization involving complex numbers.\r<BR>\n" );
document.write( "\n" );
document.write( "Given the difficulty and the lack of a clear path to the requested factorization with real numbers, it's possible that the intended factorization involves complex numbers or that the problem itself has a typo.  Without further information or context, it's difficult to proceed.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );