Factor each polynomial. See Example 7. 4(5x+7)^2+12(5x+7)+9

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor, using special products like the difference of squares, and applying the quadratic formula when necessary.

Substitution Method

The substitution method is a technique used to simplify complex polynomials by replacing a part of the expression with a single variable. In this case, substituting 'u' for '(5x + 7)' can make the polynomial easier to factor. After factoring, the original variable can be substituted back to find the complete solution.

Quadratic Form

A polynomial can often be expressed in quadratic form, which is a standard expression of the type ax^2 + bx + c. Recognizing a polynomial in this form allows for the application of various factoring techniques, such as completing the square or using the quadratic formula. Understanding this form is crucial for effectively factoring polynomials that resemble quadratics.